The study of interfacial flow is a broad topic of interest in many different research disciplines. Physicists, biologists, engineers and other scientists all share a stake in accurate representation of interfacial position. Whether pipeline flow of crude oil mixtures, cellular disruption or the development of galaxies is to be modeled, interfacial movement intimately influences calculations by accounting for a redistribution of local density functions. In general, the problems of interest have as a common element the requirement to resolve through numerical simulation the complex flow patterns that result from the flow of immiscible (or semi-immiscible) fluids.
Published works dating from the 1960's until now relate an ever-improving understanding of numerical algorithms that allow for an accurate description of interface position. These algorithms are generally categorized as either surface methods or volume methods. However, there is still no generalized method presently developed that allows for the numerical simulation of the dynamic movement of three or more fluid materials and their interfaces.
A reference field or function that moves with an interface typically characterizes surface methods. Sometimes the reference is a mass-less fixed particle along the interface (Daly, 1969; Takizawa et. al, 1992). Other times a level set function which tracks the shortest distance to an interface from a fixed point is offered (Sussman et. al, 1994; Osher and Sethian, 1988; Sethian, 1996). Further references indicate height functions are implemented where a reference line or plane is chosen in the computational domain (Nichols and Hirt, 1973). All of these methods give accurate descriptions of two fluid, single interface problems that do not involve folding, breaking or merging interfaces. The biggest advantage of surface methods is that an interface position is explicitly known for all time.
Volume methods, however, build a reference within the fluids under evaluation. Typically a method will discretely identify different materials on either side of an interface. Mixed cells then indicate the general location of an interface. An exact location is never discretely known, and volume methods are characterized by a reconstruction step whereby an approximate interface is built from local data consisting of volume and area fractions.
The volume methods may be further divided into 2 sub-cases to include particle methods and scalar advection methods. Particle methods include the marker and cell (MAC) method (Harlow and Welch, 1965; Daly, 1967) and the particle in cell (PIC) method (Harlow et.al, 1976). These algorithms employ the idea of mass-less particles to identify a particular material and then track their movements through a static grid based on local velocity conditions. The volume of fluid (VOF) method (Hirt and Nichols, 1981; Ashgriz and Poo, 1991; Lafaurie et.al, 1994) and various line techniques including SLIC (Noh and Woodward, 1976), PLIC (Youngs, 1982) and FLAIR (Ashgriz and Poo, 1991) implement several unidirectional sweeps to predict the change in volume fractions of cells at each time step based on a local resolution of the scalar transport equation. There are many combination algorithms presently used which implement both VOF and line techniques to capture even greater interface detail (Rider and Kothe, 1998; Gueyffier et al., 1999; Scardovelli et al., 2000, 2002). Volume methods offer good interfacial descriptions of complicated fluid geometries in both two and three dimensions and allow for interface folding, breaking, and merging. However these methods are restricted to cases involving only two materials.
Traditional VOF methods define a concentration function, C, to denote materials. Typically,
  C  =      {                                        1            ,                                                fluid            ⁢                                                  ⁢            1                                                            0            ,                                                fluid            ⁢                                                  ⁢            2                              where values between 0 and 1 represent mixed cells.
C is then transported by the velocity field u via the scalar transport equation,
                    ∂        C                    ∂        t              +          u      ·              ∇        C              =  0.And finally, the average (or cell centered) values of density, ρ, and viscosity, μ, are interpolated as: ρ=Cρ1+(1−C)ρ2 and μ=Cμ1+(1−C)μ2 where the subscripts 1 and 2 refer to materials 1 and 2. In this interpretation, C, although called a concentration function, acts as the fluid fraction of material 1 or 2 in any given cell.
Methods have been employed previously which have attempted to track two interfaces or three materials in a computational fluid simulation. These methods (Eulerian-Eulerian) operate by using control volumes which may contain at most three materials and two unique interfaces. As discussed above, at least a single set of conservation equations is still required to resolve the flow field dynamics, i.e., mass conservation equation and two or three momentum conservation equations dependent on the dimensionality of the problem (2D or 3D), or a complete set of conservation equations is solved for each material. However, no one to date has addressed the issue of uniquely identifying and tracking, with a single or multiple concentration function(s) Ci, three or more materials. The issue always distills down to the unique tracking and resolving of multiple interfaces in a computational cell. In addition, when multiple equation sets are used as possibly in Eulerian-Eulerian methods, the numerical problem becomes intractable when folding, breaking, and reforming interfaces exist due to the need to precisely define the interface shape inorder to apply the constitutive relationships between materials in this formulation. The above described computational fluid dynamic methods fail to provide a method which can readily handle explicit or implicit tracking of the interface between three or more generally immiscible fluid materials. The present invention provides an efficient and tractable method which overcomes the deficiency of current methods in tracking multiple fluid materials and their interfaces.